well, i think the condition should be given that a, b, and c are distinct. Otherwise a=b=c=0 (considering a polynomial with no constants) satisifes. So contradicts.

 

 

 

Proof:

 

now suppose we have and then by (1) we get,

and

so and , for some integers and  

 

So Hence Thus and

 

That gives us now by (1),

but Thus , which is impossible....answer follows.